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The given series is:

\[
8 + 6 + \frac{9}{2} + \frac{27}{8} + \dots
\]

We need to determine if the series converges or diverges, and if it converges, find its sum.

### Step 1: Identify the type of series
This appears to be a geometric series. A geometric series has the form:

\[
a + ar + ar^2 + ar^3 + \dots
\]

where \( a \) is the first term, and \( r \) is the common ratio between consecutive terms. Let's check if there's a common ratio between the terms.

- First term \( a = 8 \).
- Second term \( 6 = 8 \times r \), so \( r = \frac{6}{8} = \frac{3}{4} \).
- Third term \( \frac{9}{2} = 6 \times r \), so \( r = \frac{\frac{9}{2}}{6} = \frac{9}{12} = \frac{3}{4} \).
- Fourth term \( \frac{27}{8} = \frac{9}{2} \times r \), so \( r = \frac{\frac{27}{8}}{\frac{9}{2}} = \frac{27}{8} \times \frac{2}{9} = \frac{3}{4} \).

Thus, the common ratio \( r = \frac{3}{4} \), confirming that this is a geometric series.

### Step 2: Test for convergence
A geometric series converges if the absolute value of the common ratio \( |r| < 1 \). In this case:

\[
\left| \frac{3}{4} \right| = \frac{3}{4} < 1
\]

So, the series converges.

### Step 3: Find the sum of the series
The sum \( S \) of an infinite geometric series is given by the formula:

\[
S = \frac{a}{1 - r}
\]

where \( a \) is the first term and \( r \) is the common ratio. Substituting the values:

\[
S = \frac{8}{1 - \frac{3}{4}} = \frac{8}{\frac{1}{4}} = 8 \times 4 = 32
\]

Thus, the sum of the series is **32**.

---

Would you like more details or have any questions? Here are some related questions to explore:

1. How do you test for convergence in other types of series?
2. What happens to a geometric series when the ratio \( r \) is greater than or equal to 1?
3. Can a geometric series have negative terms and still converge?
4. How is the sum formula for a finite geometric series derived?
5. What are some real-world applications of geometric series?

**Tip:** In a geometric series, when \( |r| < 1 \), each successive term becomes smaller and smaller, ensuring the series converges.

Determine if the given series converge or diverge. If the series converges, find the sum: 8 + 6 + 9/2 + 27/8 + ...

Explore the convergence of the geometric series 8 + 6 + 9/2 + 27/8 + ... and learn how to find its sum. This detailed solution covers essential concepts like series convergence, geometric series, and the formula for the sum of an infinite geometric series. Suitable for students in Grades 11-12.

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